I'm considering my own rel@tionship with my sister here and how this would work out. I would be more than happy to offer her accommodations in my home, but she would have to understand that I and my family will not be around much as we already have other plans. We will not be available to entertain, drive, or babysit. Here's the key, make yourself at home. This would work with my sister, who probably already has needs met for transportation and babysitting and anything else necessary for travel, she just needs a roof, hopefully for free. Hopefully her and I can bump into each other and have a little time together in the interim before we rush off to our obligations. It's a perfectly workable plan.
It doesn't sound like your sister is that entirely reliable in this arena.
Who the hell calls at 6 o'clock in the morning? If you're too fuzzy-brained to function at that time, in the future, you need to simply say you'll call them back. At 6 a.m., if you are up at that time, you're probably getting ready for work or getting the kids ready for school, and it's really not a good time to be dealing with anything un-emergent anyway...you can talk later, most things can wait. It also gives you some time polish your spine and put some thoughts together on this impending commitment.
Putting sister up for a night or two is no big deal, in my opinion, if she can otherwise fend for herself, but if you're expected to drive her everywhere, babysit, and provide entertainment at the expense of your own plans, the answer should have been no, and you're not rude to say no, and if they call at the butt-crack of dawn you need to say you'll call back later.
Your mom is making her own choices in this whirlwind of last minute plans. This is not your responsibility, mom can say no too. You are not contributing. That aspect is independent of your choices. Your mom can stress or not stress, you really have no control here.